1. Field
Example embodiments relate to methods for evaluating robustness of solutions to constraint problems. Also, example embodiments relate to methods for evaluating robustness of solutions to constraint problems including core design, operational strategy, or core design and operational strategy for nuclear reactors.
2. Description of Related Art
Most problems encountered in engineering design are nonlinear by nature and involve the determination of system parameters that satisfy certain goals for the problem being solved. Such problems can be cast in the form of a mathematical optimization problem where a solution is desired that maximizes and/or minimizes one or more system functions or parameters subject to one or more limitations or constraints on the system. Both the system function and constraints are comprised of system inputs (control variables) and system outputs, which may be either discrete or continuous. Furthermore, constraints may be equalities or inequalities. The solution to a given optimization problem has either or both of the following characteristics: (1) minimizes and/or maximizes a desired condition or conditions, thus satisfying the optimality condition; and (2) satisfies a set of constraint equations imposed on the system.
With the above definitions, several categories of optimization problems may be defined. A Free Optimization Problem (“FOP”) is one for which no constraints exist. A Constraint Optimization Problem (“COP”) includes both constraints and minimize and/or maximize one or more conditions requirement. In contrast, a Constraint Satisfaction Problem (“CSP”) contains only constraints. Solving a CSP means finding one feasible solution within the search space that satisfies the constraint condition(s). Solving a COP means finding a solution that is both feasible and optimal in the sense that the minimum and/or maximum value(s) for the one or more desired conditions is/are realized.
The solution to such a problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, hopefully results in improvement of an objective function. An objective function is a mathematical expression having parameter values of a proposed solution as inputs. The objective function produces a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another. Numerous search algorithms exist and differ in the manner by which the control variables for a particular problem are modified, whether a population of solutions or a single solution is tracked during the improvement process, and the assessment of convergence. However, these search algorithms rely on the results of an objective function in deciding a path of convergence. Examples of optimization algorithms include Genetic Algorithms, Simulated Annealing, and Tabu Search.
Within optimization algorithms, the issue of handling constraints for COPs and CSPs must be addressed. Several classes of methods exist for dealing with constraints. The most widespread method is the use of the penalty approach for modifying the objective function, which has the effect of converting a COP or CSP into a FOP. In this method, a penalty function, representing violations in the set of constraint equations, is added to an objective function characterizing the desired optimal condition. When the penalty function is positive, the solution is infeasible. When the penalty function is zero, all constraints are satisfied. Minimizing the modified objective function thus seeks not only optimality but also satisfaction of the constraints.
For a given optimization search, the penalty approach broadens the search space by allowing examination of both feasible and infeasible solutions in an unbiased manner. Broadening the search space during an optimization search often allows local minima to be circumnavigated more readily, thus making for a more effective optimization algorithm. In contrast, alternate methods for handling constraints, such as infeasible solution “repair” and “behavioral memory,” are based on maintaining or forcing feasibility among solutions that are examined during the optimization search.
To implement the penalty approach, a mathematical expression is defined for each constraint that quantifies the magnitude of the constraint violation. For the given constraint, a weighting factor then multiplies the result to create an objective function penalty component. Summing all penalty components yields the total penalty. The larger the weighting factor for a given constraint, the greater the emphasis the optimization search will place on resolving violations in the constraint during the optimization search. Many approaches exist for defining the form of the penalty function and the weighting factors. As defined by the resultant modified objective function, weighting factors are problem specific and are bounded by zero (the constraint is not active) and infinity (the search space omits all violations of the constraint).
The simplest penalty function form is the “death penalty,” which sets the value of the weighting factor for each constraint to infinity. With a death penalty the search algorithm will immediately reject any violation of a constraint, which is equivalent to rejecting all infeasible solutions. Static penalties apply a finite penalty value to each constraint defined. A static weighting factor maintains its initial input value throughout the optimization search. Dynamic penalties adjust the initial input value during the course of the optimization search according to a mathematical expression that determines the amount and frequency of the weight change. The form of the penalty functions in a dynamic penalty scheme contains, in addition to the initial static penalty weighting factors (required to start the search), additional parameters that must be input as part of the optimization algorithm.
Similar to dynamic penalties, adaptive penalties adjust weight values over the course of an optimization search. In contrast, the amount and frequency of the weight change is determined by the progress of the optimization search in finding improved solutions. Several approaches for implementing adaptive penalty functions have been proposed. Bean and Hadj-Alouane created the method of Adaptive Penalties (“AP”), which was implemented in the context of a Genetic Algorithm. In the AP method, the population of solutions obtained over a preset number of iterations of the optimization search is examined and the weights adjusted depending on whether the population contains only feasible, infeasible, or a mixture of feasible and infeasible solutions. Coit, Smith, and Tate proposed an adaptive penalty method based on estimating a “Near Feasibility Threshold” (“NFT”) for each given constraint. Conceptually, the NFT defines a region of infeasible search space just outside of feasibility that the optimization search would then be permitted to explore. Eiben and Hemert developed the Stepwise Adaption of Weights (“SAW”) method for adapting penalties. In their method, a weighting factor adjustment is made periodically to each constraint that violates in the best solution, thus potentially biasing future solutions away from constraint violations.
Several deficiencies exist in the penalty methods proposed. Death penalties restrict the search space by forcing all candidate solutions generated during the search to satisfy feasibility. In the static weighting factor approach, one must perform parametric studies on a set of test problems that are reflective of the types of optimization applications one would expect to encounter, with the result being a range of acceptable weight values established for each constraint of interest. The user would then select the weight values for a specific set of constraints based on a pre-established range of acceptable values. Particularly for COPs, varying the static weight values for a given problem can often result in a more or less optimal result. Similarly, dynamic penalties require the specification of parameters that must be determined based on empirical data. Fine-tuning of such parameters will often result in a different optimal result.
Penalty adaptation improves over the static and dynamic penalty approaches by attempting to utilize information about the specific problem being solved as the optimization search progresses. In effect, the problem is periodically redefined. A deficiency with the adaptive penalty approach is that the objective function loses all meaning in an absolute sense during the course of an optimization search. In other words, there is no “memory” that ties the objective function back to the original starting point of the optimization search as exists in a static penalty or dynamic penalty approach.
One known optimization problem involves design of an operational strategy for a nuclear reactor such as a nuclear boiling water reactor (“BWR”). FIG. 1 illustrates a related art BWR. As shown, a pump 100 supplies water to a reactor vessel 102 housed within a containment vessel 104. The core 106 of the reactor vessel 102 includes a number of fuel bundles such as those described in detail below with respect to FIG. 2. The controlled nuclear fission taking place at the fuel bundles in the core 106 generates heat which turns the supplied water into steam. This steam is supplied from the reactor vessel 102 to turbines 108, which power a generator 110. The generator 110 then outputs electrical energy. The steam supplied to the turbines 108 is recycled by condensing the steam from turbines 108 back into water at a condenser 112, and supplying the condensed steam back to the pump 100.
FIG. 2 illustrates a typical fuel bundle 114 in the core 106. A core 106 may include, for example, anywhere from about 200 to about 1200 of these fuel bundles 114. As shown in FIG. 2, the fuel bundle 114 may include an outer channel 116 surrounding a plurality of fuel rods 118 extending generally parallel to one another between upper and lower tie plates 120 and 122, respectively, and in a generally rectilinear matrix of fuel rods as illustrated in FIG. 3, which is a schematic representation of a cross-section or lattice of the fuel bundle 114 of FIG. 2. The fuel rods 118 may be maintained laterally spaced from one another by a plurality of spacers 124 vertically spaced apart from each other along the length of the fuel rods 118 within the outer channel 116. Referring to FIG. 3, there is illustrated in an array of fuel rods 118 (i.e., in this instance, a 10×10 array) surrounded by the outer channel 116. The fuel rods 118 are arranged in orthogonally related rows and also surround one or more “water rods,” two water rods 126 being illustrated. The fuel bundle 114 may be arranged, for example, in one quadrant of a control blade 128 (also known as a “control rod”). It will be appreciated that other fuel bundles 114 may be arranged in each of the other quadrants of the control blade 128. Movement of the control blade 128 up and/or down between the fuel bundles 114 controls the amount of reactivity occurring in the fuel bundles 114 associated with that control blade 128.
A nuclear reactor core includes many individual components that have different characteristics that may affect a strategy for efficient operation of the core. For example, a nuclear reactor core has many (i.e., several hundred) individual fuel assemblies (bundles) that have different characteristics and that must be arranged within the reactor core (or “loaded”) so that the interaction between the fuel bundles satisfies all regulatory and reactor design constraints, including governmental and customer-specified constraints. Similarly, other controllable elements and factors that affect the reactivity and overall efficiency of a reactor core must also be taken into consideration if one is to design or develop an effective strategy for optimizing the performance of a reactor core at a particular reactor plant. Such “operational controls” (also referred to interchangeably in this application as “independent control-variables” and “design inputs”) include, for example, various physical component configurations and controllable operating conditions that may be individually adjusted or set.
Besides fuel bundle “loading,” other independent control-variables include, for example, “core flow” or rate of water flow through the core, the “exposure,” and the “reactivity” or interaction between fuel bundles within the core due to differences in bundle enrichment, and the “rod pattern” or distribution and axial position of control blades within the core. As such, each of these operational controls constitutes an independent control-variable or design input that has a measurable effect on the overall performance of the reactor core. Due to the vast number of possible different operational values and combinations of values that these independent control-variables can assume, it is a formidable challenge and a very time consuming task, even using known computer-aided methodologies, to attempt to analyze and optimize all the individual influences on core reactivity and performance.
For example, the number of different fuel bundle configurations possible in the reactor core may be in excess of one hundred factorial. Of the many different loading pattern possibilities, only a small percentage of these configurations may satisfy all of the requisite design constraints for a particular reactor plant. In addition, only a small percentage of the configurations that satisfy all of the applicable design constraints may be economically feasible.
Moreover, in addition to satisfying various design constraints, since a fuel bundle loading arrangement ultimately affects the core cycle energy (i.e., the amount of energy that the reactor core generates before the core needs to be refueled with fresh fuel bundles), a particular loading arrangement should to be selected that may optimize the core cycle energy.
In order to furnish and maintain the required energy output, the reactor core is periodically refueled with fresh fuel bundles. The duration between one refueling and the next is commonly referred to as a “fuel cycle” or “core cycle” of operation and, depending on the particular reactor plant, is on the order of twelve to twenty-four (typically eighteen) months. At the time of refueling, typically one third of the least reactive fuel bundles are removed from the reactor and the remaining fuel bundles are repositioned before fresh fuel bundles are added. Generally, to improve core cycle energy, higher reactivity bundles should be positioned at interior core locations. However, such arrangements are not always possible to achieve while still satisfying plant-specific design constraints. Since each fuel bundle may be loaded at a variety of different locations relative to other bundles, identifying a core loading arrangement that produces optimum performance of the core for each fuel cycle presents a complex and computation-intensive optimization problem that may be very time consuming to solve.
During the course of a fuel cycle, the excess energy capability of the core, defined as the excess reactivity or “hot excess,” is controlled in several ways. One technique employs a burnable reactivity inhibitor (i.e., Gadolinia) incorporated into the fresh fuel. The quantity of initial burnable inhibitor is determined by design constraints and performance characteristics typically set by the utility and by the Nuclear Regulatory Commission (“NRC”). The burnable inhibitor controls most, but not all, of the excess reactivity. Consequently, control blades—that inhibit reactivity by absorbing nuclear emissions—are also used to control excess reactivity. Typically, a reactor core contains many such control blades that are fitted between selected fuel bundles and are axially positionable within the core. These control blades assure safe shut down and provide the primary mechanism for controlling a maximum core power peaking factor.
The total number of control blades utilized varies with core size and geometry, and may be, for example, between about 50 and about 175. The axial position of the control blades (i.e., fully inserted, fully withdrawn, or somewhere in between) is based on the need to control the excess reactivity and to meet other operational constraints, such as the maximum core power peaking factor. For each control blade, there may be, for example, 24, 48, or more possible axial positions or “notches” and 40 “exposure” (i.e., duration of use) steps. Considering symmetry and other requirements that reduce the number of control blades that are available for application at any given time, there are many millions of possible combinations of control blade positions for even the simplest case. Of these possible configurations, only a small fraction may satisfy all applicable design and safety constraints, and of these, only a small fraction may be economically feasible. Moreover, the axial positioning of control blades also influences the core cycle energy that any given fuel loading pattern can achieve. Since it is desirable to maximize the core cycle energy in order to minimize nuclear fuel cycle costs, developing an optimum control blade positioning strategy presents another formidable independent control-variable optimization problem that must also be taken into consideration when attempting to optimize fuel-cycle design and management strategies.
Control blades are typically grouped and assigned a designation, such as ‘A1’, ‘A2,’ ‘B1,’ and ‘B2’. Only the control blades within a specified group may be used for control of the reactor over a designated period of time. For example, dividing the core into 8 exposure periods (i.e., time periods) of 2 months each, a typical operational strategy might be the ordered use of blades within the following groups: ‘B1,’ ‘A1,’ ‘B2,’ ‘A2,’ ‘B1,’ ‘A1,’ ‘B2,’ ‘A2’ The time boundary between any two groups is called a sequence exchange, such as ‘B1 ’→‘A1’ which occurs at the completion of the first 2 month period. Within a group, individual control blades are placed at notch positions, which correspond to a certain fraction of insertion. For example, notch 48 corresponds to completely withdrawn while notch 0 corresponds to completely inserted. Symmetric blades may be ganged and will therefore move in unison. Typically symmetries are octant, quarter-core (mirror or rotational), and half-core rotational.
Control blades are moved to control local power within the reactor core as the fuel depletes as well as to control the reactivity of the core. In conjunction with control blades, core flow may also be used as a control mechanism. The higher the core flow, the more the core reactivity and vice-versa. Similarly, the deeper a control blade is inserted, the lower the core reactivity and vice-versa. The impact of a given blade on core reactivity and local power depends on a number factors including: (1) the location of the blade—blades near the core periphery in low power regions have less of an impact than those in higher power regions such as near the center; (2) the characteristics of the fuel bundles surrounding the blade (i.e., fresh fuel or highly exposed fuel); (3) the number of symmetric partners (a ganging of 8 control blades has greater impact than a ganging of 4 control blades); (4) the core exposure; and (5) the current core state power distribution (inserting a blade for an axially bottom-peak power shape will have greater impact than for a top-peaked power shape).
Core design and the development of an operational strategy typically involves a constraint optimization problem wherein a best possible solution that maximizes energy output is developed according to various well-known algorithms. For example, a reactor core and operational strategy may be designed to generate a certain amount of energy measured in gigawatt days per metric ton (“GWd/mt”) of uranium over a fuel cycle before being replaced with a new core.
As discussed above, developing a solution to such a constraint problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, hopefully results in improvement of an objective function, producing a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another. Numerous search algorithms for core and operational strategy design exist that rely on the results of an objective function in deciding a path of convergence.
At the beginning of cycle (“BOC”), the core design is put into operation. As is also typical, actual reactor performance often deviates from the performance modeled in generating the core design. Adjustments from the operational model are quite often made in order to maintain performance of the reactor before the end of cycle (“EOC”). Accordingly, the desire for robustness in a design solution arises from the fact that the assumptions that form the basis of a given design may change once the plant starts operating. Assumptions fall into several categories. First, there are assumed operational conditions of the plant that include, for example, the power level, flow, and inlet temperature. Second, there are assumed biases in the simulation model that are based on historical data. As is known, developing a core and/or operational strategy design solution involves running simulations of the reactor using a proposed solution and using outputs from the simulation as inputs to an objective function, which provides a figure of merit for the propose solution. Numerous simulation programs for simulating reactor performance are known in the art. An example of a simulation model bias is the core eigenvalue, which is a measure of core reactivity or neutron balance, at hot and cold conditions as function of cycle exposure (for a critical core the eigenvalue should be 1.00 but typically ranges between 0.99 and 1.01).
Another category of assumption is assumed margins in the simulation model for each of the thermal and reactivity parameters. Design margins are introduced to account for uncertainties in the simulation model and to assure that once the plant starts operating, thermal and reactivity limits are not violated (the so-called operating margin). Examples of reactivity limits include cold shutdown margin (“CSDM”) and hot excess reactivity (“HOTX”). Examples of thermal limits include MFLPD (Maximum Fraction of Limiting Power Density), MAPRAT (maximum power ratio), and MFLCPR (Maximum Fraction of Limiting Critical Power Ratio).
CSDM is defined as the reactivity margin to the limit for the reactor in a cold state, with all control blades inserted except for the most reactive control blade. CSDM may be determined for each exposure state-point during the fuel cycle. HOTX is defined as the core reactivity for the reactor in a hot state, with all control blades removed. Similar to CSDM, HOTX may be determined for each exposure state-point during the fuel cycle.
MFLPD is defined as the maximum of the ratio of local rod power or linear heat generation rate (“LHGR”; measured, for example, in kilowatts per unit length) in a given bundle at a given elevation, as compared to its limiting value. MAPLHGR is the maximum average linear heat generation rate over a plane in a given bundle at a given elevation in the core. MAPRAT is simply the ratio of MAPLHGR to its limiting value.
LHGR limits protect the fuel against the phenomena of fuel cladding plastic strain, fuel pellet centerline melting, and lift-off (bulging of the cladding that exceeds the expansion of the pellet, due primarily to fission gas buildup). Lift-off degrades heat transfer from the pellet across the cladding to the coolant. MAPRAT limits protect the fuel during a postulated loss of coolant accident (“LOCA”), while MFLPD limits protect the fuel during normal operation.
MFLCPR protects the fuel against the phenomena of “film dryout.” In BWR heat transfer, a thin film of water on the surface of the fuel rod assures adequate removal of the heat generated in the fuel rod as water is converted into steam. This mechanism, also known as nucleate boiling, will continue as the power in the fuel rod is increased up until a point referred to as transition boiling (also known as departure from nucleate boiling or “DNB”). During transition boiling, heat transfer degrades rapidly leading to the elimination of the thin film and ultimately film dryout, at which time the cladding surface temperature increases rapidly, possibly leading to cladding failure. The critical power of the bundle is the power at which a given fuel bundle achieves film dryout, and is determined from experimental tests. The Critical Power Ratio (“CPR”) is the ratio of the critical power to the actual bundle power. MFLCPR is simply the maximum over all bundles of the fraction of each bundle's CPR to its limiting value.
Operating margins may be communicated to a core monitoring system, and are derived from plant measurement or instrumentation systems. In a BWR, the instrumentation systems may include fixed detectors and removable detectors. The removable detectors, or TIPS (traversing in-core probes), are inserted each month to calibrate the fixed detectors. This is due to the fact that the fixed detectors will “burn out” due to the neutron environment and so must have their signals adjusted. As will be appreciated, however, in a simulator the measurements are simulated. A loss of operating margin may require adjustment of the control blade pattern and/or core flow in order to redistribute the power. The control blade pattern is the amount by which each of the control blades is inserted into the core and how these positions are planned to change over time. Core flow is the flow of water through the core.
Changes in any of the design assumptions—operational conditions, model biases, or margins—may require changes in the reactor control parameters once the plant begins operation. Avoidance of abrupt changes in core output response (i.e., local power) due to a required change in one of the independent control-variables (i.e., axial position of one or more control blades) is important from the perspective of plant safety as well as ease of operation.
Core design is currently performed using a fixed set of assumptions. This method of design does not provide information as to the robustness of a given solution. A design may satisfy all design margins for the input set of assumption but may prove to have reduced margins (or worse, approach violations in thermal and/or reactivity limits) during plant operation. In such instances, the reactor operators would modify the operational strategy (control blade pattern and/or core flow) to recover at least some of the lost margin. Typically, such modifications to the operational strategy would be first simulated using the on-line predictive capabilities of the core monitoring system, beginning with a “snapshot” of the plant state based on the plant measurement and operating conditions. During the simulation of these various scenarios, the degree of robustness of the current solution will become evident. Robust solutions should have low sensitivity to modifications in operational strategy. A solution that is brittle may require additional operational maneuvering (such as using an alternate group of control blades) in order to achieve a robust solution. This maneuvering may require a reduction in core power (and lost electrical generation) during the “transition” maneuver to the new core state.
An alternative method is to perform a simulation of the base design with a single change in one of the design parameters and validate that a success path—involving a change in operational strategy—exists for satisfying the thermal and reactivity limits. For example, one could change the target hot eigenvalue from 1.000 to 1.003 over the fuel cycle and manually perturb control blades and core flow within the simulation to satisfy thermal and reactivity limits. If no such success path existed, it would be necessary to change the overall design. Examples of such changes would be to perform fuel shuffles, utilize a different set of control blades (i.e., an A1 sequence versus an A2 sequence), or modify the fresh fuel bundle design. This process is extremely time consuming and can only examine singular changes in the design parameters.